Characterization of Conditional Independence and Weak Realizations of Multivariate Gaussian Random Variables: Applications to Networks

The Gray and Wyner lossy source coding for a simple network for sources that generate a tuple of jointly Gaussian random variables (RVs) $X_1 : Omega rightarrow {mathbb R}^{p_1}$ and $X_2 : Omega rightarrow {mathbb R}^{p_2}$, with respect to square-error distortion at the two decoders is re-examined using (1) Hotellings geometric approach of Gaussian RVs-the canonical variable form, and (2) van Puttens and van Schuppens parametrization of joint distributions ${bf P}_{X_1, X_2, W}$ by Gaussian RVs $W : Omega rightarrow {mathbb R}^n $ which make $(X_1,X_2)$ conditionally independent, and the weak stochastic realization of $(X_1, X_2)$. Item (2) is used to parametrize the lossy rate region of the Gray and Wyner source coding problem for joint decoding with mean-square error distortions ${bf E}big{||X_i-hat{X}_i||_{{mathbb R}^{p_i}}^2 big}leq Delta_i in [0,infty], i=1,2$, by the covariance matrix of RV $W$. From this then follows Wyners common information $C_W(X_1,X_2)$ (information definition) is achieved by $W$ with identity covariance matrix, while a formula for Wyners lossy common information (operational definition) is derived, given by $C_{WL}(X_1,X_2)=C_W(X_1,X_2) = frac{1}{2} sum_{j=1}^n ln left( frac{1+d_j}{1-d_j} right),$ for the distortion region $ 0leq Delta_1 leq sum_{j=1}^n(1-d_j)$, $0leq Delta_2 leq sum_{j=1}^n(1-d_j)$, and where $1 > d_1 geq d_2 geq ldots geq d_n>0$ in $(0,1)$ are {em the canonical correlation coefficients} computed from the canonical variable form of the tuple $(X_1, X_2)$. The methods are of fundamental importance to other problems of multi-user communication, where conditional independence is imposed as a constraint.